#137043

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂×#41	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6,9,12)(3,10,11,13)(17,18,21,26,22,27,36,44,23,28,37,45,30,39,20,25)(19,29,32,42,33,43,31,46,34,47,35,24,38,40,48,41);(1,2,7,6,5,9,4,12)(3,8,10,14,11,16,13,15)(17,19,30,33,23,34,22,38)(18,24,39,41,28,42,27,46)(20,35,37,48,36,32,21,31)(25,43,45,47,44,40,26,29);(1,3)(2,8)(4,13)(5,11)(6,14)(7,10)(9,16)(12,15)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44239);